Fundamental matrix (linear differential equation)

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations$$ \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) $$is a matrix-valued function $$ \Psi(t) $$ whose columns are linearly independent solutions of the system. Then every solution to the system can be written as $$\mathbf{x}(t) = \Psi(t) \mathbf{c}$$, for some constant vector $$\mathbf{c}$$ (written as a column vector of height $n$).

A matrix-valued function $$ \Psi $$ is a fundamental matrix of $$ \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) $$ if and only if $$ \dot{\Psi}(t) = A(t) \Psi(t) $$ and $$ \Psi $$ is a non-singular matrix for all $ t $.

Control theory
The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations.